Bernoulli--Strang--Fix Conditions: Approximation and Prediction by Sampling Kantorovich Operators

Abstract

In this paper, we introduce the Bernoulli--Strang--Fix conditions and their generalized versions for a vector-valued generator $\phi =({\phi }_0,\dots ,{\phi }_{\rho -1})$ and a periodic nonuniform sampling set $X$. We use these conditions to establish exact and asymptotic polynomial reproduction properties of sampling Kantorovich operators associated with $(\phi ,X)$ up to a prescribed degree. We analyze the approximation properties and convergence behavior of these operators in detail. Furthermore, we demonstrate their application to signal prediction from a finite number of past local average samples, showing that sampling Kantorovich operators can also serve as effective prediction operators. Finally, we present numerical examples based on Gaussian functions and B-splines to illustrate and validate the theoretical approximation and prediction results.

20 pages, 3 figures